Stability Properties of Systems of Linear Stochastic Differential Equations with Random Coefficients

Bishop, Adrian N.; Moral, Pierre Del

doi:10.1137/18m1182759

Cited by 13 publications

(14 citation statements)

References 62 publications

(164 reference statements)

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“…Additional results are applicable if we restrict κ = 0. We have the following immediate corollary of our prior work in [9]:…”

Section: Contraction Properties Of Exponential Semigroupsmentioning

confidence: 72%

“…Several local-type contraction estimates can be derived. For instance, the stochastic semigroup E ǫ s,t (Q) exhibits the following stability properties derived as immediate corollaries of our work in [9]:…”

Section: Contraction Properties Of Exponential Semigroupsmentioning

confidence: 77%

“…The stability properties of stochastic semigroups associated with a collection of stochastic flows (A, A ǫ ) satisfying the above properties have been developed in our prior work [9]. Several local-type contraction estimates can be derived.…”

Section: Contraction Properties Of Exponential Semigroupsmentioning

confidence: 99%

“…whenever ǫ ≤ ε n,t where here ε n,t is the largest parameter ǫ such that t ǫ n > t n ; see [9] for more details on these time parameters.…”

Section: Contraction Properties Of Exponential Semigroupsmentioning

confidence: 99%

“…via the connection in (3.9). Indeed, from our prior work in [9], see Section 2.2.1, we have the following stability estimates:…”

Section: Ensemble Filtering Stability Propertiesmentioning

confidence: 99%

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On the stability of matrix-valued Riccati diffusions

Bishop¹,

Moral²

2019

Electron. J. Probab.

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The stability properties of matrix-valued Riccati diffusions are investigated. The matrixvalued Riccati diffusion processes considered in this work are of interest in their own right, as a rather prototypical model of a matrix-valued quadratic stochastic process. In addition, this class of stochastic models arise in signal processing and data assimilation, and more particularly in ensemble Kalman-Bucy filtering theory. In this context, the Riccati diffusion represents the flow of the sample covariance matrices associated with McKean-Vlasov-type interacting Kalman-Bucy filters. Under rather natural observability and controllability conditions, we derive time-uniform moment and fluctuation estimates and exponential contraction inequalities. Our approach combines spectral theory with nonlinear semigroup methods and stochastic matrix calculus. The analysis developed here applies to filtering problems with unstable signals. This analysis seem to be the first of its kind for this class of matrix-valued stochastic differential equation.1/2This special case (κ = 0) defines, in some sense, a minimal prototype of a forward-in-time matrixvalued Riccati diffusion in the space of symmetric positive (semi-)definite matrices. We let φ ǫ t (Q) := Q t be the stochastic flow of the matrix diffusion equation (1.2). Whenever it exists, the inverse stochastic flow of (1.2) is denoted by φ −ǫ t (Q) := Q −1 t . For any 0 ≤ s ≤ t, we let E ǫ s,t (Q) be the transition semigroup associated with the flow of random matrices [A − φ ǫ t (Q)S], i.e. the solution of the forward and backward equationsfor any 0 ≤ s ≤ t, with E ǫ t,t (Q) = I. When s = 0 we write E ǫ t (Q) instead of E ǫ 0,t (Q). We write φ t (Q) and E s,t (Q) instead of φ 0 t (Q), and E 0 s,t (Q), to denote the flow of the deterministic matrix Riccati differential equation when ǫ = 0, and the exponential semigroup defined via φ t (Q). 2

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“…Additional results are applicable if we restrict κ = 0. We have the following immediate corollary of our prior work in [9]:…”

Section: Contraction Properties Of Exponential Semigroupsmentioning

confidence: 72%

Section: Contraction Properties Of Exponential Semigroupsmentioning

confidence: 77%

Section: Contraction Properties Of Exponential Semigroupsmentioning

confidence: 99%

“…whenever ǫ ≤ ε n,t where here ε n,t is the largest parameter ǫ such that t ǫ n > t n ; see [9] for more details on these time parameters.…”

Section: Contraction Properties Of Exponential Semigroupsmentioning

confidence: 99%

“…via the connection in (3.9). Indeed, from our prior work in [9], see Section 2.2.1, we have the following stability estimates:…”

Section: Ensemble Filtering Stability Propertiesmentioning

confidence: 99%

See 3 more Smart Citations

On the stability of matrix-valued Riccati diffusions

Bishop¹,

Moral²

2019

Electron. J. Probab.

Self Cite

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A New Approach to Solve Linear Fuzzy Stochastic Differential Equation

Panda

Dash²,

Panda³

2022

Meta Heuristic Techniques in Software Engineering and Its Applications

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On the mathematical theory of ensemble (linear-Gaussian) Kalman–Bucy filtering

Bishop

Moral

2023

Math. Control Signals Syst.

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The purpose of this review is to present a comprehensive overview of the theory of ensemble Kalman–Bucy filtering for continuous-time, linear-Gaussian signal and observation models. We present a system of equations that describe the flow of individual particles and the flow of the sample covariance and the sample mean in continuous-time ensemble filtering. We consider these equations and their characteristics in a number of popular ensemble Kalman filtering variants. Given these equations, we study their asymptotic convergence to the optimal Bayesian filter. We also study in detail some non-asymptotic time-uniform fluctuation, stability, and contraction results on the sample covariance and sample mean (or sample error track). We focus on testable signal/observation model conditions, and we accommodate fully unstable (latent) signal models. We discuss the relevance and importance of these results in characterising the filter’s behaviour, e.g. it is signal tracking performance, and we contrast these results with those in classical studies of stability in Kalman–Bucy filtering. We also provide a novel (and negative) result proving that the bootstrap particle filter cannot track even the most basic unstable latent signal, in contrast with the ensemble Kalman filter (and the optimal filter). We provide intuition for how the main results extend to nonlinear signal models and comment on their consequence on some typical filter behaviours seen in practice, e.g. catastrophic divergence.

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Stability Properties of Systems of Linear Stochastic Differential Equations with Random Coefficients

Cited by 13 publications

References 62 publications

On the stability of matrix-valued Riccati diffusions

On the stability of matrix-valued Riccati diffusions

A New Approach to Solve Linear Fuzzy Stochastic Differential Equation

On the mathematical theory of ensemble (linear-Gaussian) Kalman–Bucy filtering

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